Question

Hello, I have a differential equation question !! any helper here?

Answer 1

\[\frac{ dy }{ dx } - \frac{ 2 }{x } y = x^4 \sin(x) \]

Answer 2

looks like linear d.e.

Answer 3

yes?

Answer 4

the question is to solve the above differential equation brother

Answer 5

yes...i was saying that the d.e. is linear...

Answer 6

\[\frac {dx}{dy} + P(x) y = Q(x)\]
do you remember that?

Answer 7

I have started already but not sure I am on the right truck or not

Answer 8

wait....have you learned about linear differential equations yet?

Answer 9

I learned how to apply integration on this , where I put the intergal on each side and trake y to left and x to the right.

Answer 10

hmm you're talking about the method of variable separation..

Answer 11

okay...let me see if i can turn this into that

Answer 12

hmm...are you sure you're supposed to solve this differential equation already? i mean...if you haven't learned the method of linear differential equations then i don't think this is possible to solve...

or...it could be there's a way to use the method you're describing...but i need to be sure that it's possible...

so...are you sure you are supposed to solve this already?

Answer 13

we do differetial equation by seprable.

Answer 14

Is this what you mean ?

Answer 15

no...i know your lesson is variable separable....my question is...are you really supposed to solve this problem? because i have doubts that it can be solved by variable separable...but if you're really supposed to solve this using that method....theni can probably try to find a way to solve this using that method

Answer 16

check the file please

Answer 17

hmm and?

Answer 18

by the way...that pdf file mentions the method of variable separable and linear d.e. (which was what i was mentioning earlier)

Answer 19

this is my homework, lool

Answer 20

so i suppose...you're supposed to know the method of first order linear d.e.

Answer 21

ok
If possible could you teach me please

Answer 22

I did the first one using by parts but now I am working on the second

Answer 23

the method? or how to solve this using that method?

Answer 24

anything you like I want to learn

Answer 25

i can't teach you the method....but i can teach you how to solve this....just refer to your pdf file as your guide....don't worry...it won't be too complicated

Answer 26

okay... so you have \[\huge \frac{dy}{dx} - \frac 2x y = x^4 \sin x\]

first step:

integrate -2/x y

what do you get?

Answer 27

typo... integrate -2/x

what do you get?

Answer 28

^integrate -2/x dx

Answer 29

ok

Answer 30

what do you get?

Answer 31

before we start i want to show what I did

Answer 32

sure

Answer 33

can I ?

Answer 34

that would help me in determining where you're stuck

Answer 35

\[-\int\limits_{}^{} dy=\int\limits_{}^{}x^4 \sin(x)dx \]

Answer 36

no... you can't solve this using variable separable

Answer 37

ok thanks for that

Answer 38

let's start then

Answer 39

look at your pdf file #1 c...that's the method we're using

Answer 40

first step is determining the integrating factor

the integrating factor can be solved using this formula \[I.F. = e^{\int P(x)dx}\]given\[\frac{dy}{dx} + P(x) y = Q(x)\]

in our equation, we have \[\frac{dy}{dx} - \frac 2x y = x^4 \sin x\]
our P(x) in this case is -2/x

do you follow so far?

Answer 41

yep with you

Answer 42

good. so our integrating factor would be \[\huge I.F. = e^{\int \frac 2x dx}\]
can you solve this?

Answer 43

wait...typo \[\huge I.F. = e^{\int -\frac 2x dx}\]

Answer 44

I could not

Answer 45

do you know how to solve this integral? \[\huge \int -\frac 2x dx\]

Answer 46

I know the rule of integration e but with power integration never did

Answer 47

ignore the e first....concentrate on that integral...do you know how to solve that integral?

Answer 48

2/x^2

Answer 49

not exactly...that was the derivative

Answer 50

let me help you \[\huge \int -\frac 2x dx \implies -2\int \frac 1xdx\]
do you know the integral of 1/x?

Answer 51

integral x is one

Answer 52

integral of x is not one....that's the derivative.....

Answer 53

and i'm asking integral of 1/x

Answer 54

ln (x)

Answer 55

right!

Answer 56

\[\int -\frac 2x dx \implies -2\ln x\]

Answer 57

now...here comes some algebra...do you agree with this:
\[-2\ln x \implies \ln x^{-2}\]

Answer 58

yep I agree

Answer 59

no need for ln I reckon

Answer 60

so \[\large e^{\int -\frac 2x dx} \implies e^{-2 \ln x} \implies e^{\ln x^{-2}}\]
yes?

Answer 61

yep

Answer 62

do you know how to simplify \[\Large e^{\ln (x^{-2})}\]

Answer 63

not really

Answer 64

here's a hint: \[\large e^{\ln a} \implies a\]

Answer 65

I messaged you check please

Answer 66

i replied

Answer 67

Yes this is MathDude000. I was wondering if someone can help me here. please reply back if you are busy. I will find someone else to study if you are busy

Answer 68

hello?

Answer 69

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Answer 70

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